Endoscopic contribution conjecture for Siegel modular forms of genus two

Let $(l,m)$ be regular. The endoscopic contribution $e_{\rm endo}({\mathcal A}_2,\mathbb V_{l,m})$ is defined using the cohomology of the moduli space ${\mathcal A}_2$ of principally polarized abelian surfaces with coefficient system $\mathbb V_{l,m}$; write $s_n=\dim S_n(\Gamma_1)$ and let $S[\,\cdot\,]$ denote the corresponding motive or cohomological contribution of elliptic cusp forms. Endoscopic contribution conjecture.

$$e_{\rm endo}({\mathcal A}_2,\mathbb V_{l,m})=-s_{l+m+4}S[l-m+2] \,\mathbb L^{m+1}.$$

This predicts the part of the interior cohomology arising from endoscopic lifting from $N={\rm GL}(2)\times {\rm GL}(2)/{\mathbb G}_m$; it is based on numerical calculations, and the source gives no resolution status.